For the purposes of testing and checking rotary bodies and more especially rotary bodies of an annular configuration, of a flexible material, such as for example a motor vehicle tire, it is necessary for the rotary body to be arranged in the testing machine in a centered position to carry out the quality control operation. For example, quality control operations are carried out on motor vehicle tires in equipment referred to as tire uniformity measuring machines. In such a machine, a pneumatic tire to be measured and checked is mounted in position on measuring rim portions, and rolled against a test surface. During that operation, measurements are made of irregularities or non-uniformities of the forces involved, in order to arrive at an indication of the quality of the tire. For that purpose, it is necessary for the clamping devices, for holding the tire in position in the machine, to have a centering relationship. A similar consideration applies if the machine is so designed as to mount a wheel with a tire thereon, for carrying out a quality control operation on the tire. In practice it has been found that, particularly when the centering arrangement also provides for measuring the diameter of the rotary body to be subjected to the testing operation, incorrect measurements may occur when dealing with different successive types of tires, so that in that case the clamping devices in the machine for holding the tire or the vehicle wheel in the appropriate position therein are not disposed in a properly centralizing relationship.